File size: 2,644 Bytes
c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | import OAI.Probability.InvariantIsing.Fields.FieldEndpointProduct
import OAI.Probability.InvariantIsing.Fields.FieldEndpointMoment
namespace OAI
/-! Integrability of linear-growth vector observables under the actual
canonical endpoint law, uniformly allowing zero-variance increments. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
lemma integrable_linearGrowth_pi {N : ℕ} (ν : Fin N → Measure ℝ)
[∀ i, IsProbabilityMeasure (ν i)] (hν : ∀ i, Integrable (fun x : ℝ => x) (ν i))
(F : (Fin N → ℝ) → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F) :
Integrable F (Measure.pi ν) := by
obtain ⟨C, L, _, hL, hb⟩ := hG
have hi (i : Fin N) : Integrable (fun z : Fin N → ℝ => |z i|) (Measure.pi ν) := by
exact (measurePreserving_eval ν i).integrable_comp_of_integrable (hν i).abs
have hsum := integrable_finsetSum Finset.univ (fun i _ => hi i)
apply ((integrable_const C).add (hsum.const_mul L)).mono' hF.aestronglyMeasurable
apply ae_of_all
intro z
rw [Real.norm_eq_abs]
apply (hb z).trans
apply add_le_add le_rfl
apply mul_le_mul_of_nonneg_left _ hL
apply (pi_norm_le_iff_of_nonneg (Finset.sum_nonneg (fun i _ => abs_nonneg (z i)))).mpr
intro i
simpa only [Real.norm_eq_abs] using
(Finset.single_le_sum (fun j _ => abs_nonneg (z j)) (Finset.mem_univ i))
lemma fieldTailEndpointKernel_integrable_id (L : List (ℝ × ℝ≥0))
(hL : ∀ av ∈ L, 0 < av.1) (hB : ∀ av ∈ L, av.1 ≤ 1) (z : ℝ) :
Integrable (fun y : ℝ => y) (fieldTailEndpointKernel L hL z) := by
have hp := (fieldTailEndpointKernel_mgf L hL hB 1 z).1
have hn := (fieldTailEndpointKernel_mgf L hL hB (-1) z).1
apply (hp.add hn).mono' measurable_id.aestronglyMeasurable
apply ae_of_all
intro y
rw [Real.norm_eq_abs]
simp only [one_mul, neg_one_mul, Pi.add_apply, id_eq]
by_cases hy : 0 ≤ y
· rw [abs_of_nonneg hy]
linarith [Real.add_one_le_exp y, Real.exp_pos (-y)]
· rw [abs_of_nonpos (le_of_not_ge hy)]
linarith [Real.add_one_le_exp (-y), Real.exp_pos y]
lemma fieldVectorTailEndpointKernel_integrable (N : ℕ) (L : List (ℝ × ℝ≥0))
(hL : ∀ av ∈ L, 0 < av.1) (hB : ∀ av ∈ L, av.1 ≤ 1)
(z : Fin N → ℝ) (F : (Fin N → ℝ) → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F) :
Integrable F (fieldVectorTailEndpointKernel N L hL z) := by
rw [fieldVectorTailEndpointKernel_product]
exact integrable_linearGrowth_pi _ (fun i => fieldTailEndpointKernel_integrable_id L hL hB (z i))
F hF hG
end InvariantIsing
end
end OAI
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